The space L1∩L∞

Ben Zion A. Rubshtein; Genady Ya Grabarnik; Mustafa A. Muratov; Yulia S. Pashkova

doi:10.1007/978-3-319-42758-4_3

The space L₁∩L_∞

Ben Zion A. Rubshtein, Genady Ya Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

The space L₁ ∩ L_∞, which we study in this chapter, consists of all bounded integrable functions equipped with the norm ∥·∥_{L₁ ∩ L∞}= max(∥·∥L₁, ∥·∥_L∞). We show that. (L₁ ∩ L_∞; ∥·∥_{L₁ ∩ L∞}) is a symmetric space and describe the closure L⁰_∞ of L₁ ∩ L_∞ in L_∞. Given two equimeasurable functions f and g, we treat an approximation of g in the L₁ ∩ L_∞-norm by shifted functions f o θ, where θ is a measure-preserving transformation. Step functions and integrable simple functions are applied for this purpose.

Original language	English
Title of host publication	Developments in Mathematics
Publisher	Springer New York LLC
Pages	29-40
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-319-42758-4_3
State	Published - 1 Jan 2016

Publication series

Name	Developments in Mathematics
Volume	45
ISSN (Print)	1389-2177

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-319-42758-4_3

Cite this

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abstract = "The space L1 ∩ L∞, which we study in this chapter, consists of all bounded integrable functions equipped with the norm ∥·∥L1 ∩ L∞= max(∥·∥L1, ∥·∥L∞). We show that. (L1 ∩ L∞; ∥·∥L1 ∩ L∞) is a symmetric space and describe the closure L0∞ of L1 ∩ L∞ in L∞. Given two equimeasurable functions f and g, we treat an approximation of g in the L1 ∩ L∞-norm by shifted functions f o θ, where θ is a measure-preserving transformation. Step functions and integrable simple functions are applied for this purpose.",

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